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Determination of the $ D\sp {1/2}$-norm of the SOR iterative matrix for the unsymmetric case

Authors: D. J. Evans and C. Li
Journal: Math. Comp. 53 (1989), 203-218
MSC: Primary 65F10; Secondary 65N99
MathSciNet review: 969486
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Abstract: This paper is concerned with the determination of the Jordan canonical form and $ {D^{1/2}}$-norm of the SOR iterative matrix derived from the coefficient matrix A having the form

$\displaystyle A = \left( {\begin{array}{*{20}{c}} {D1} \hfill & { - H} \hfill \\ {{H^T}} \hfill & {{D_2}} \hfill \\ \end{array} } \right)$

with $ {D_1}$ and $ {D_2}$ symmetric and positive definite. The theoretical results show that the Jordan form is not diagonal, but has only q principal vectors of grade 2 and that the $ {D^{1/2}}$-norm of $ {\mathcal{L}_{{\omega _b}}}$ ( $ {\omega _b}$, the optimum parameter) is less than unity if and only if $ \bar \mu = \rho (B)$, the spectral radius of the associated Jacobi iterative matrix, is less than unity. Here q is the multiplicity of the eigenvalue $ i\bar \mu $ of B.

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Keywords: SOR iterative matrix, $ {D^{1/2}}$-norm, spectral radius, spectral norm and Jordan canonical form
Article copyright: © Copyright 1989 American Mathematical Society